Added the source of the updated Mandic trapping model

The trapping probability is calculated as an exponential decay as a function of the simulation timestep as

```math

p_{e, h} = \left(1 - \exp^{1 \frac{\delta t}{\tau_{e, h}}}\right)

p_{e, h} = \left(1 - \exp^{

    1 \frac{\delta t} {

        \tau_ { e, h }

    }}\right)

```

where $`\delta t`$ is the simulation timestep and $`\tau{e,h}`$ the effective lifetime of electrons and holes, respectively.

At the same time, a total time spent in the trap is calculated if a detrapping model is selected. Here, the time until the

charge carrier is de-trapped is calculated as

where $`\delta t`$ is the simulation timestep and $`\tau{

    e, h

}

`$ the effective lifetime of electrons and holes, respectively.At the same time,

    a total time spent in the trap is calculated if a detrapping model is selected.Here,

    the time until the charge carrier is de - trapped is calculated as

```math

\delta t = - \tau_{e.h} \ln{1-p}

\delta t = - \tau_ {

    e.h

}

\ln { 1 - p }

```

where $`p`$ is a probability randomly chosen from a uniform distribution between 0 and 1.

In the Ljubljana (sometimes referred to as *Kramberger*) model \[[@kramberger]\], the trapping time follows the relation

```math

\tau^{-1}(T) = \beta(T)\Phi_{eq} ,

\tau^{

    -1}(T) = \beta(T)\Phi_{eq} ,

```

where the temperature scaling of $`\beta`$ is given as

```math

\beta(T) = \beta(T_0)\left(\frac{T}{T_0}\right)^{\kappa} ,

\beta(T) = \beta(T_0)\left(\frac{T}{

    T_0}\right)^{

    \kappa} ,

```

extracted at the reference temperature of $`T_0 = -10 \,\text{°C}`$.

extracted at the reference temperature of $`T_0 = -10 \,\text{

    °C

}

`$.

    The parameters used in Allpix Squared are

```math

\begin{aligned}

\begin {

    aligned

}

\beta_{e}(T_0) &= 5.6\times 10 ^ { -16 } \,\text{cm} ^ 2\,\text{ns} ^ { -1 } \\

\kappa_{e} &= -0.86 \\

\\

\beta_{h}(T_0) &= 7.7\times 10 ^ { -16 } \,\text{cm} ^ 2\,\text{ns} ^ { -1 } \\

\kappa_{h} &= -1.52

\end{aligned}

\end {

    aligned

}

```

for electrons and holes, respectively.

for protons have been applied here.

The parameters arise from measurements of the were obtained evaluating current signals of irradiated sensors via light

injection at fluences up to $`\Phi_{eq} = 2\times 10^{14} \ n_{eq}\,\text{cm}^2`$.

injection at fluences up to $`\Phi_{eq} = 2\times 10^{

    14} \ n_{

    eq

}

\,\text{cm} ^

      2`$.

This model can be selected in the configuration file via the parameter `trapping_model = "ljubljana"`.

      This model can be selected in the configuration file via the parameter `trapping_model =

      "ljubljana"`.

      ## #Dortmund

The Dortmund (sometimes referred to as *Krasel*) model \[[@dortmundTrapping]\], describes the effective trapping times as

      The

      Dortmund(sometimes referred to as * Krasel*) model \[[@dortmundTrapping]\],

                                                                             describes the effective trapping times as

```math

\tau ^ { -1 } = \gamma\Phi_{eq},

    with the parameters

```math

\begin{aligned}

\begin {

    aligned

}

\gamma_{e} &= 5.13\times 10 ^ { -16 } \,\text{cm} ^ 2\,\text{ns} ^ { -1 } \\

\gamma_{h} &= 5.04\times 10 ^ { -16 } \,\text{cm} ^ 2\,\text{ns} ^ { -1 }

\end{aligned}

\end {

    aligned

}

```

for electrons and holes, respectively.

The values have been extracted evaluating current signals of irradiated sensors via light injection at fluences up to

$`\Phi_{eq} = 8.9 \times 10^{14}\ n_{eq}\,\text{cm}^2`$, at a temperature of $`T = 0\,\text{°C}`$. No temperature scaling is

$`\Phi_{eq} = 8.9 \times 10^{

    14}\ n_{

    eq

}

\,\text{cm} ^ 2`$, at a temperature of $`T = 0\,\text {°C }`$. No temperature scaling is

provided. Values for neutron and proton irradiation have been evaluated in \[[@dortmundTrapping]\], Allpix Squared makes use

of the values for proton irradiation.

### CMS Tracker

This effective trapping model has been developed by the CMS Tracker Group. It follows the results of

\[[@CMSTrackerTrapping]\], with measurements at fluences of up to $`\Phi_{eq} = 3 \times 10^{15} \ n_{eq}\,\text{cm}^2`$, at

a temperature of $`T = -20 \,\text{°C}`$ and an irradiation with protons.

\[[@CMSTrackerTrapping]\], with measurements at fluences of up to $`\Phi_{eq} = 3 \times 10^{

    15} \ n_{

    eq

}

\,\text{cm} ^ 2`$, at a temperature of $`T = -20 \,\text {°C }

`$ and an irradiation with protons.

    The interpolation of the results follows the relation

```math

\tau^{-1} = {\beta\Phi_{eq}} + \tau_0^{-1}

\tau ^

{ -1 } = {\beta\Phi_{eq}} + \tau_0 ^ { -1 }

```

         with the parameters

```math

\begin{aligned}

\begin {

    aligned

}

\beta_{e}(T_0)  &= 1.71\times 10^{-16} \,\text{cm}^2\,\text{ns}^{-1} \\

\tau_{0,e}^{-1} &= -0.114 \,\text{ns}^{-1} \\

\\

\beta_{h}(T_0)  &= 2.79\times 10^{-16} \,\text{cm}^2\,\text{ns}^{-1} \\

\tau_{0,h}^{-1} &= -0.093 \,\text{ns}^{-1}

\end{aligned}

\end{

    aligned

}

```

for electrons and holes, respectively.

### Mandic

The Mandić model \[[@Mandic]\] is an empirical model developed from measurements with high fluences ranging from

$`\Phi_{eq} = 5\times 10^{15} \ n_{eq}\,\text{cm}^2`$ to $`\Phi_{eq} = 1\times 10^{17} \ n_{eq}\,\text{cm}^2`$ and describes

$`\Phi_{eq} = 5\times 10^{

    15} \ n_{

    eq

}

\,\text{cm} ^ 2`$ to $`\Phi_{eq} = 1\times 10 ^ { 17 } \ n_ { eq }

\,\text{cm}^2`$ and describes

the lifetime via

```math

```math

\begin{aligned}

c_e      &= 0.54 \,\text{ns}\,\text{cm}^{-2} \\

c_e      &= 0.54 \,\text{

    ns

}

\,\text{cm} ^ { -2 } \\

\kappa_e &= -0.62 \\

\\

c_h      &= 0.0427 \,\text{ns}\,\text{cm}^{-2} \\

\ c_h &= 0.0427 \,\text {

    ns

}

\,\text{cm} ^ { -2 } \\

\kappa_h &= -0.62

\end{aligned}

\end {

    aligned

}

```

for electrons and holes, respectively.

The parameters for electrons are taken from \[[@Mandic]\], for measurements at a temperature of $`T = -20 \,\text{°C}`$, and

the results extrapolated to $`T = -30 \,\text{°C}`$. Note that an erratum has been made (see https://iopscience.iop.org/article/10.1088/1748-0221/16/03/E03001), c_e should be 0.54ns instead of 0.054ns.

The parameters for electrons are taken from \[[@Mandic]\], for measurements at a temperature of $`T = -20 \,\text{

    °C

}

`$, and the results extrapolated to $`T = -30 \,\text {°C }`$. 

Note that an erratum has been made, see [@MandicErratum]: https://doi.org/10.1088/1748-0221/16/03/E03001, the c_e should be 0.54ns instead of 0.054ns.

A scaling from electrons to holes was performed based on the default values in Weightfield2 \[[@weightfield2]\].

This model can be selected in the configuration file via the parameter `trapping_model = "mandic"`.

are denoted with squared brackets and a parameter number, for example `[0]` for the first parameter provided. Parameters

specified separately from the formula can contain units which will be interpreted automatically.

{{% alert title="Note" color="info" %}}

{

    { % alert title = "Note" color = "info" % }}

Both fluence and temperature are not inherently available in the custom trapping model, but need to be provided as additional

parameters as described above.

{{% /alert %}}

{

    { % / alert % }}

The following configuration parameters replicate the [Ljubljana model](#ljubljana) using a custom trapping model.

     * @brief Mandic effective trapping model

     *

     * Parametrization taken from https://doi.org/10.1088/1748-0221/15/11/P11018, section 5.

     * The c_e should be 0.54ns instead of 0.054ns, see the erratum from Prof. Mandić https://iopscience.iop.org/article/10.1088/1748-0221/16/03/E03001.

     * Update c_e to 0.54ns, see [@MandicErratum]: https://doi.org/10.1088/1748-0221/16/03/E03001.

     * Scaling from electrons to holes taken from default beta values in Weightfield2

     */

    class Mandic : virtual public TrappingModel {